School

Seattle University **We aren't endorsed by this school

Course

CHEM 455

Subject

Mathematics

Date

Oct 12, 2023

Pages

2

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1
THE UNCERTAINTY PRINCIPLE (1927)
Werner Heisenberg
- Nobel Prize in Physics (1932) at age of 31!
Mentor: Niels Bohr
The values
of par�cular pairs of observables cannot be determined
simultaneously with arbitrarily high precision in quantum mechanics.
Examples of pairs of observables that are restricted in this way are momentum
and posi�on, and energy and �me; such pairs are referred to as
complementary.
The quan�ta�ve expressions of the Heisenberg uncertainty principle can be
derived by combining the de Broglie rela�on
p
=
h
/
λ
and the Einstein rela�on
E
=
hv
with proper�es of all waves.
The de Broglie wave for a par�cle is made up of a superposi�on of an infinitely
large number of waves of the form
(
)
(
)
,
sin 2
sin 2
x
x t
A
vt
A
x
vt
ψ
π
λ
π κ
=
−
=
−
where
A
is amplitude and
κ
is the reciprocal
wavelength.
We consider one spa�al dimension for simplicity. The
waves that are added together have infinitesimally
different wavelengths. This superposi�on of waves
produces a
wave packet
shown to the right.
1
1
4
x
x
κ
λ
π
∆
∆
=∆
∆
≥
(1)
(a) weakly localized
1
4
t
v
π
∆ ∆
≥
(2)
(b) strongly localized
where Δ
x
is the extent of the wave packet in space, Δ
κ
is the range in reciprocal
wavelength, Δ
v
is the range in frequency, and Δ
t
is a measure of the �me required
for the packet to pass a given point. The Δ's in these equa�ons are actually
standard devia�ons.

2
If at a given �me the wave packet extends over a short range of
x
values, there is a
limit to the accuracy with which we can measure the wavelength. If a wave packet
is of short dura�on, there is a limit to the accuracy with which we can measure
the frequency.
S
ubs�tu�ng the de Broglie rela�on in equa�on
(1)
,
1/λ =
p
x
/
h
for mo�on in the
x
direc�on, then
1
4
x
p
x
h
π
∆
∆
≥
2
x
x
p
∆
∆
≥
ħ
=
h
/2
π
(
h
bar)
Another form of the uncertainty principle may be derived by subs�tu�ng
E
=
hv
in
equa�on (2) and it
yields
1
4
E
t
h
π
∆ ∆
≥
2
t
E
∆ ∆
≥

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